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Deeper than primes

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nathan said:
huh? I thought everybody thought in images. I'm always visualizing and drawing diagrams of maths, code, data, whatever. Doesn't prevent me explaining stuff to people though.
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Good.

So please tell me what prevents from you to immediately understand the simple notion that an object like a line segment is not determined by points?
 
doronshadmi said:
So please tell me what prevents from you to immediately understand the simple notion that an object like a line segment is not determined by points?
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You don't understand what the sentence, "Two distinct points determine a line", means, do you?
 
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doronshadmi said:
I understand and disagree with it becuse no finitely or infinitely many points (where each point is local by nature) can determine a line segment (which is non-local by nature).

The reason is clearly seen in http://www.geocities.com/complementarytheory/UR.pdf .
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But you are not disagreeing with what you quoted. What you quoted means something else entirely and you are poving that you do not understand what it is saying.

You are so hell bent on making your point you don't even realize, or even care, that you talking about something different from what you quoted.
 
doronshadmi said:
Please give an example of such a shortcut, taken from http://www.geocities.com/complementarytheory/UR.pdf .
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Consider Definition #3:

Set A is called non-finite if the cardinal of proper subset B of A can be |B| <= |A|.​

Probably not what you were expecting as a shortcut, but it represents a shortcut, nonetheless. Doron, you rush so quickly to throw out words, symbols, and pictures, you must not realize the result doesn't mean at all what your want it to mean.

By this definition, all non-empty sets are non-finite.
 
jsfisher said:
Consider Definition #3:

Set A is called non-finite if the cardinal of proper subset B of A can be |B| <= |A|.​

Probably not what you were expecting as a shortcut, but it represents a shortcut, nonetheless. Doron, you rush so quickly to throw out words, symbols, and pictures, you must not realize the result doesn't mean at all what your want it to mean.

By this definition, all non-empty sets are non-finite.
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I agree with you. Thank you.


The idea is to say that, not like definition 2 (where the cardinal of proper subset B of A cannot be but |B| < |A|) in definition 3 the cardinal of proper subset B of A is at least |B| < or = |A|.

Let us fix it to:

Definition 3: Set A is called non-finite if the cardinal of proper subset B of A is at least |B| < or = |A|.
 
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doronshadmi said:
I agree with you. Thank you.


The idea is to say that, not like definition 2 (where the cardinal of proper subset B of A cannot be but |B| < |A|) in definition 3 the cardinal of proper subset B of A is at least |B| < or = |A|.

Let us fix it to:

Definition 3: Set A is called non-finite if the cardinal of proper subset B of A is at least |B| < or = |A|.
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Are you alleging a difference between the relation, "<=", and your shorthand notation, "< or ="? I see no change in meaning in your proposed "fix". Nor do I have any idea what utility you believe the phrase "at least" added to the definition.

By the way, your Definition #2 is also faulty. Be that as it may, though, are you totally unaware that if you successfully define "Property X", it then is unnecessary to define "Property not-X" (aka "Property non-X")?
 
doronshadmi said:
Good.

So please tell me what prevents from you to immediately understand the simple notion that an object like a line segment is not determined by points?
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Well done for changing the subject. Bravo for not addressing the issue.
 
jsfisher said:
Are you alleging a difference between the relation, "<=", and your shorthand notation, "< or ="? I see no change in meaning in your proposed "fix". Nor do I have any idea what utility you believe the phrase "at least" added to the definition.

By the way, your Definition #2 is also faulty. Be that as it may, though, are you totally unaware that if you successfully define "Property X", it then is unnecessary to define "Property not-X" (aka "Property non-X")?
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Definition 2: Set A is called finite if the cardinal of proper subset B of A cannot be but |B| < |A|.

Definition 2 says that A is finite if its proper subset cannot be but |B| < |A|

What is not clear here? I do not think that by definition 2 we can conclude that property not-X is non-finite.

Let us try this version of definition 3:

Definition 3: Set A is called non-finite if the cardinal of proper subset B of A can be (|B| < |A|) AND (|B| = |A|).
 
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doronshadmi said:
Definition 2: Set A is called finite if the cardinal of proper subset B of A
cannot be but |B| < |A|.

Definition 2 says that A is finite if its proper subset cannot be but |B| < |A|

What is not cleat here?
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I didn't say it was uncleat unclear; I said it was faulty. You exclude at least one perfectly ordinary finite set.

Let us try this version of definition 3:

Definition 3: Set A is called non-finite if the cardinal of proper subset B of A can be (|B| < |A|) AND (|B| = |A|).
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This is worse. Are we now to the point were you tell me it is all my fault for not understanding what you mean?
 
jsfisher said:
I didn't say it was uncleat unclear; I said it was faulty. You exclude at least one perfectly ordinary finite set.



This is worse. Are we now to the point were you tell me it is all my fault for not understanding what you mean?
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Forget about the definitions in their current states.

The idea is this:


If a proper subset B of set A cannot be but |B| < |A|, then A is a finite set.

Please write definition 2 in your style.


If a proper subset B of set A can be |B| = |A| in addition to |B| < |A|, then A is a non-finite set.

Please write definition 3 in your style.


If you think that only a one definition is needed in order to distinguish between finite and non-finite sets, then please write the definition.

Thank you.
 
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doronshadmi said:
Forget about the definitions in their current states.

The idea is this:


If a proper subset B of set A cannot be but |B| < |A|, then A is a finite set.

Please write definition 2 in your style.
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"Cannot be but...." You really like that phrase, don't you? Have you ever considered expressing things in the positive rather than the negative?

That question is rhetorical, by the way. Please just consider it, but don't answer it. The real question is this:

Why are you bothering to define finite and infinite sets at all? Does it come up later in the paper?
 
jsfisher said:
"Cannot be but...." You really like that phrase, don't you? Have you ever considered expressing things in the positive rather than the negative?
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"Cannot be but ..." is an understatement of "must be ..." and I prefer to use understatements.
jsfisher said:
The real question is this:

Why are you bothering to define finite and infinite sets at all? Does it come up later in the paper?
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It will be come later.

At this stage I wish to clearly distinguish between finite and non-finite sets.

Please answer to http://www.internationalskeptics.com/forums/showpost.php?p=4236183&postcount=896 .
 
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doronshadmi said:
Forget about the definitions in their current states.
The idea is this:
If a proper subset B of set A cannot be but |B| < |A|, then A is a finite set.

Please write definition 2 in your style.


If a proper subset B of set A can be |B| = |A| in addition to |B| < |A|, then A is a non-finite set.

Please write definition 3 in your style.


If you think that only a one definition is needed in order to distinguish between finite and non-finite sets, then please write the definition.

Thank you.
Click to expand...

Without knowing what definition #1 is, I'll take a stab at definition #2.


I will be using the following definitions:
  • CardinalityWP: "In mathematics, the cardinality of a set is a measure of the "number of elements of the set". For example, the set A = {1, 2, 3} contains 3 elements, and therefore A has a cardinality of 3." Using the example set shown, I will use the notation |A| = 3 .
  • SubsetWP: "In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. Notice that A and B may coincide."
  • Infinite: something that has no end, something that cannot have a value attached to, unmeasurable/uncountable, theoretical, unquantifiable. Example, the smallest/largest number; in geometry, a line.
  • Finite: something that does have an end, can have a value attached to, measurable/countable, practical, quantifiable. Example: the number of coins in my pocket, the number of grains of sand on a beach; in geometry, a line segment.
(My rough framework of doronshamdi's "definition #2")
  • Let A be a set.
  • Set B is a subset of A.
  • If the cardinality of B is less than the cardinality of set A, then set A is a finite set.
OR
​
  • |B| < |A|, then A is finite set.
(/end #2)



But I can immediatly call BS on definition #2 using the following example:
  1. Set X is the set of all positive whole even numbers.
  2. Subset Y is the set of all positive whole even numbers divisible by five.
  3. Using just a small sample of the members of X (2,4,6,8,10), you will see that Y will only have one member.
  4. The definition of subset (see above) states that Y will be included in X.
  5. In fact, even though |Y| < |X|, X is still infinite because I did not limit the members of X.
(My rough framework of doronshamdi's "definition #3")

  1. Let A be a set.
  2. Set B is a subset of A.
  • If the cardinality of subset B is equal to the cardinality of set A, then set A is a finite set.
  • If the cardinality of subset B is also less than the cardinality of set A, then set A is a finite set.
Combining the two statements you can get:
​
  • If the cardinality of subset B is less than and equal to the cardinality of set A, then set A is a infinite set.
OR
​
  • |B| =< |A|, then A is an infinite set.
(/end #3)

Again, I call BS. If |B| < |A|, (meeting both #2 and #3) how can it be both infinite and non-infinite? In addition, how can a value be equal and greater to another value at the same time?

And if you're "At this stage I wish to clearly distinguish between finite and non-finite sets", why don't you give us your definitions of finite and non-finite sets? If you want to use your Local/Non-local relations, make sure you define them first. You know, like people have been asking you to. I'll now go back to my seat in the peanut gallery viewing section of this train wreck.
 
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